Building a wall doesn't seem very intelligent in my opinion.
There will be illegal immigrants as long as poor Mexicans have less opportunities in Mexico. And I'm pretty sure Americans know that. So why haven't the almighty USA been trying to impose changes in Mexico or working together with Mexico to really solve the roots of the problem?
This is a huge topic and likely way too broad of a topic for a simple Q/A site. But, some potential reasons to get you started:
US Trade Policies have contributed to the economic problems Mexico has had. For instance, NAFTA damaged a huge sector of Mexico's agricultural economy.
Many of the people entering the US are escaping more than economic troubles but also drug cartels--many outside of Mexico into Latin and South America. Many would argue that this is due in no small part to US policies on drugs over the past 35 years.
The US economy depends on illegal immigrants--namely migrant ag workers so there isn't necessarily universal will to do anything about it.
The honest answer would be that it isn't in their best interest for that to happen. The USA has gone as far as conspiring, via their Ambassador Henry Lane Wilson, with Huerta to kill Madero in the days of the revolution. So they are not shy of intervening in Mexico, the latest has been in the form of funding to the government of Mexico's Drug War via the Mérida Initiative.
In a comment, you say
I find this interesting because I believed that if the US really wanted to stop illegal immigration of Mexicans I'd have seen at least one news about Americans helping to combat whatever the problem is in Mexico that leads citizens to leave it.
I've seen such news shows.
The United States in Mexico, supporting Mexican police against the drug cartels.
The United States and Mexico signing NAFTA, to increase employment opportunities in Mexico.
The United States helping Mexico secure its Southern border to prevent immigration by people trying to travel through Mexico.
There will be illegal immigrants as long as poor Mexicans keep having children carelessly and as long as there are no opportunities for them in Mexico.
Is that the primary problem? Some believe that a border wall is necessary not to prevent people looking for opportunity, who are better targeted with systems like E-Verify, but to prevent reentry by criminals. Currently, drug traffickers and gang members can cross the border with relative ease. Even after being deported, they repeatedly return.
The US and Mexico have a closed border that can only be crossed legally at restricted points that require ID checks. But enforcement of that border still does not prevent people illegally in the US after multiple deportations from committing crimes.
Enforcing the border is a necessary part of law enforcement. Sticking our heads in the sand won't change that. The US has been waging war on poverty for fifty years. Hundreds of billions of dollars later, we have as much poverty now as when we started. Such programs won't magically work in Mexico where they failed in the US.
Another issue is that there are certain jobs that require transient labor to perform. Primarily related to crop harvesting, these would ideally take advantage of laborers from Mexico resting between the planting and harvesting cycles there. But under the current system, such temporary workers have too much trouble getting visas. So they cross the border illegally. And then they stay, because it is difficult to cross the border twice (leaving and then coming back next year).
Nothing that the US could do in Mexico would change that. That would have to be changed either by easier temporary visas or by finding domestic sources of labor (e.g. the inner cities, which often have high unemployment). Currently most city residents aren't interested in such jobs. And temporary visas are too difficult to get. There's too much paperwork and approval is too spotty. Illegal immigrants require no paperwork or approval, just cash.
